By the integral test, we can say that series [tex]\sum_{1}^{\infty }n^{-13}[/tex] is convergent.
Given series is [tex]\sum_{1}^{\infty }n^{-13}[/tex]
As per the integral test of convergence:
A series [tex]\sum_{1}^{\infty }n^{-13}[/tex]is convergent only if [tex]\int\limits^\infty_1 f(x)dx = \int\limits^\infty_1 {x^{-13} } \, dx[/tex] converges.
[tex]\int\limits^\infty_1 f(x)dx = [\frac{x^{-12} }{-12} ]^\infty_1[/tex]
[tex]\int\limits^\infty_1 f(x)dx=\frac{1}{12}[/tex]
Since [tex]\int\limits^\infty_1 f(x)dx[/tex] is finite i.e. 1/12 so integral converges.
So, the series [tex]\sum_{1}^{\infty }n^{-13}[/tex] also converges.
Therefore, By the integral test, we can say that series [tex]\sum_{1}^{\infty }n^{-13}[/tex] is convergent.
To get more about the convergence of series visit:
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